entropy

Article

Classification of Fractal Signals Using Two-ParameterNon-Extensive Wavelet Entropy

Julio César Ramírez-Pacheco 1,*, Joel Antonio Trejo-Sánchez 2, Joaquin Cortez-González 3

and Ramón R. Palacio 4

1 Department of Basic Sciences and Engineering, DCBeI, University of Caribe, SM-78, Mza-1, Lote-1,Esquina Fraccionamiento Tabachines, 77528 Cancún, Mexico

2 CONACYT-Centro de Investigación en Matemáticas, Carretera Sierra Papacal, Chuburna Puerto km 5,97302 Mérida, Mexico; joel.trejo@cimat.mx

3 Department of Electrical Engineering, Instituto Tecnológico de Sonora, 5 de Febrero, 818 Sur, Colonia Centro,85000 Ciudad Obregón, Mexico; joaquin.cortez@itson.edu.mx

4 Unidad Navojoa, Instituto Tecnológico de Sonora, Ramón Corona y Aguacalientes S/N, Col. ITSON,85860 Navojoa, Mexico; ramon.palacio@itson.edu.mx

* Correspondence: jramirez@ucaribe.edu.mx or cramirez@gdl.cinvestav.mx; Tel.: +52-998-881-4400 (ext. 1281)

Academic Editor: Carlo CattaniReceived: 3 March 2017; Accepted: 9 May 2017; Published: 15 May 2017

Abstract: This article proposes a methodology for the classification of fractal signals as stationary ornonstationary. The methodology is based on the theoretical behavior of two-parameter wavelet entropyof fractal signals. The wavelet (q, q′)-entropy is a wavelet-based extension of the (q, q′)-entropyof Borges and is based on the entropy planes for various q and q′; it is theoretically shownthat it constitutes an efficient and effective technique for fractal signal classification. Moreover,the second parameter q′ provides further analysis flexibility and robustness in the sense that different(q, q′) pairs can analyze the same phenomena and increase the range of dispersion of entropies.A comparison study against the standard signal summation conversion technique shows that theproposed methodology is not only comparable in accuracy but also more computationally efficient.The application of the proposed methodology to physiological and financial time series is alsopresented along with the classification of these as stationary or nonstationary.

Keywords: fractal signal classification; fractional Gaussian noise/fractional Brownian motion(fGn/fBm) dichotomy; wavelets; non-extensive entropies; two-parameter entropies

1. Introduction

Fractal signals are ubiquitous in nature [1–6] and many aspects of these processes appear inhardly any field of science [3,4,7–10]. An important research topic within the fractal theory concernsthe estimation of fractal signals [5,6,11,12], specifically, the fractality parameter. Within fractal signalanalysis and estimation, fractal signal classification (i.e., the classification of signals as stationaryor nonstationary) plays an important role since it determines the type of estimator to be used,the form of important statistics such as correlations and moments and also the smoothness of samplepaths [7,10,12–14]. Some authors have even claimed that the signal classification should be consideredas the primary step for fractal signal analysis [13] and have devised methodologies for the classificationof fractal processes. Many techniques have attempted to classify fractal signals as stationary ornonstationary. The signal summation conversion (SSC) [7,13] is a technique based on the observedbehavior of the power spectral density (PSD) estimator when estimating the fractality parameter in therange α ∈ (−1, 3). The PSD underestimates fractal signals in a range of the fractality exponent andtherefore an additional technique is used instead (scaled window variance (SWV)). SSC is therefore a

Entropy 2017, 19, 224; doi:10.3390/e19050224 www.mdpi.com/journal/entropy

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PSD-based technique and is described in detail in [7,13]. The fractal signal classification scheme basedon wavelet Tsallis q-entropy was first proposed in [14] and takes advantage of the particular behavior ofentropies (nonextensive Tsallis) for fractal signals. Both techniques, however, are subject to inaccuraciesin the stationary/nonstationary limit (α → 1) but unlike the SSC methodology, the wavelet Tsallisq-entropy technique appears to be more computationally efficient [14], therefore it is more appropriatefor long signals. In this article, two-parameter nonextensive wavelet entropy is proposed for the fractalsignal classification problem. The concept of entropy, as applied to statistical mechanics, measuresthe information content or uncertainty of a random signal or system [15,16]. The entropy of a signal,in their diverse functionals, has been proposed to solve a handful of problems across the numerousfields of science. For instance, the well-known Shannon, Rényi and Tsallis entropies have been usedin [17–19] to assess the complexities associated to random and nonlinear phenomena. The literature hasalso witnessed the rise of more elaborate information theory functionals which consist of the productof complexity/entropy measures and allow for greater flexibility and the possibility to analyze a widerrange of phenomena. Current applications of the aforementioned entropic measures include but are notlimited to the analysis of two-electron systems [20] and their extension to many electron systems [21].Also, information planes are being applied to randomness and the localization of molecules [22] and forthe analysis of X-ray astrophysical sources [23]. With the advent of multiscale analyses, more preciseand fast information theory quantifiers have been proposed. The so-called wavelet entropy [24,25]constitutes an extension of the Shannon entropy using a probability function obtained from the waveletrepresentation of the signal. Extensions of the wavelet entropy include the wavelet q-entropy using aTsallis functional and the generalized wavelet Fisher’s information [26]. The analysis of a signal intheir wavelet domain has many advantages including the possibility to handle nonstationarities and toadjust the analyses to selective signal resolutions. Wavelet information tools have found applicationsin various fields including physiology, e.g., for electroencephalogram (EEG) and electrocardiogram(ECG) signal analysis [25,27], and physics [27], among others [28]. Wavelet information tools, thereforeembody a field that is under constant development and may offer significant tools and advantagesfor signal and system analysis. This article, as mentioned above, proposes a methodology that usesthe observed behavior of wavelet (q, q′)-entropies of fractal signals for the classification of these asstationary or nonstationary. The motivation behind the use of the wavelet (q, q′)-entropy stems fromthe fact that fractal signals experiment constant entropies over time while nonstationary fractal signalsdisplay variable entropies across time. In addition, parameter q′ provides further analysis flexibilityand the ability to use different configurations of the (q, q′) pair to analyze the same phenomena.Moreover, wavelet (q, q′)-entropy constitutes a generalization of the wavelet entropy and waveletq-entropy approaches and therefore, the results reported in the literature using these entropies may bereplicated with the wavelet (q, q′)-entropy. The rest of the article is organized as follows: Section 2provides a review of some important results regarding the wavelet analysis of fractal signals andpresents some important properties of the wavelet (q, q′)-entropy as well. The signal classificationproblem is also presented and a brief description of the techniques available for performing this isgiven. Section 3 presents some results regarding the classification of fractal signals using the wavelet(q, q′)-entropy and Section 4 presents a comparison study with the SSC using synthesized fractalsignals and the application of the proposed methodology in gait and financial time series. FinallySection 5 concludes the article.

2. Materials and methods

2.1. The Representation of Fractal Signals by Wavelets

Fractal signals are ubiquitous in nature and are representative models for the phenomenaoccurring in deoxyribonucleic acid (DNA) sequences [29], heart-beat time series [30], and moodand self-steem [31], among others. The PSD is the traditional approach used in the literature to definea fractal process. In fact, a fractal signal is defined as a process for which their PSD is:

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S( f ) ∼ c f | f |−α, f ∈ ( fa, fb), (1)

where c f is a constant, α ∈ R is the fractality parameter and fa, fb represent the lower and upperbounds upon which (1) holds [32,33]. Many stochastic processes can be used to model the power-lawbehavior of the PSD of fractal processes (as given in Equation (1)). The pure-power-law (PPL) process,the fractional auto-regressive integrated moving-average (f-ARIMA) time series and the well-knownfractional Brownian motion (fBm) and fractional Gaussian noise (fGn) are examples of fractal stochasticprocesses [34]. In this contribution, the classification of fractal signals as stationary or nonstationary isconsidered. Two types of fractal stochastic processes are studied, the fGn/fBm family and the PPLfamily of random signals. FBm is a Gaussian process which is self-similar and nonstationary [6,35,36].FBm is defined in terms of two interrelated concepts, the correlation function which is given by:

EBH(t)BH(s) =σ2

2

{|t|2H + |s|2H − |t− s|2H

}, (2)

with H ∈ (0, 1) and spectral density function (SDF) which satisfies the following relation:

S f Bm( f ) ∼ c| f |−(2H+1), f → 0. (3)

Equation (3) implies that the fBm is a fractal signal with α = 2H + 1 [6,36]. FGn, on the otherhand, is a differenced version of fBm and is stationary, Gaussian, self-similar and has a PSD of theform [6,11,35–37]:

S f Gn = 4σ2XcH sin2(π f )

∞

∑j=−∞

1| f + j|2H+1 , | f | < 1

2, (4)

with H ∈ (0, 1). In the limit of f → 0, the PSD of fGn behaves as S f Gn ∼ c| f |−2H+1 and therefore isa fractal signal with α = 2H − 1. The PPL family of stochastic processes is defined as a process forwhich their PSD is:

SPPL( f ) = Cs| f |−α, f ≤ 1/2, (5)

for Cs > 0. When α < 1, the PPL process is stationary and when α ≥ 1, it is nonstationary. The fGn/fBmand the PPL families of processes behave similarly although they differ in their PSD. Wavelet analysis,originally proposed in geophysics [38], decomposes a signal in terms of small waves and permits tocompute the so-called multiresolution analysis [39,40]. For random signals, the wavelet spectrum [41]permits computation of many interesting quantities such as the relative wavelet energy (RWE) whichbasically represents a probability function for the signal under study [24,42]. The wavelet spectrum offractal signals was obtained in the work of Abry and Veitch [32,33] and can be computed using thefollowing relation,

Ed2X(j, k) =

∫ ∞

−∞SX(2−j f )|Ψ( f )|2 d f , (6)

where Ψ( f ) =∫

ψ(t)e−j2π f t dt is the Fourier integral of the mother wavelet ψ(t), SX(.) is the PSD ofthe process Xt, E the expectation operator and dX(j, k) is the discrete wavelet transform (DWT) of theprocess Xt at time k and wavelet scale j [32,33]. Using the PSD of fractal signals given by Equation (1)and substituting it into Equation (6) results in the wavelet spectrum of fractal signals,

Ed2X(j, k) = C2jα, (7)

where C is a constant. Equation (7) has been used for the estimation of the fractality exponent α [32]and also for computing wavelet-based information tools [14,24,25,42–44]. For further information onwavelets and in the wavelet analysis of fractal signals refer to [32,33,39,40,45–49] and references therein.

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2.2. A Nonextensive Wavelet (q, q′)-Entropy of Fractal Signals

In this article, a methodology for the classification of fractal signals based on the behavior oftheir wavelet (q, q′)-entropy is proposed. The wavelet (q, q′)-entropy is an extension (to the time-scaledomain) of the (q, q′)-entropy first proposed by Borges in [50]. In order to obtain a closed-formexpression for the wavelet (q, q′)-entropy of fractal signals, their RWE is computed. The RWE of fractalsignals was obtained in [24] using the wavelet spectrum equation of (7) and results in:

πj = 2(j−1)α 1− 2α

1− 2αN , (8)

where N ∈ Z. Note that πj in (8) for j = 1, 2, . . . , N represents a probability mass function (pmf) in thesense that πj ∈ (0, 1), ∑j πj = 1 and therefore many information theory quantifiers can be obtained forfractal signals using this relation. The application of (8) into the definition of (q, q′)-entropy [50] andgiven by the relation,

HT(q,q′)(πj) =

N

∑j=1

πqj − π

q′j

N1−q − N1−q′ , (9)

results in the wavelet (q, q′)-entropy of fractal signals which is given by,

HT(q,q′)(πj) =

{(1−2α

1−2αN

)q ( 1−2αqN

1−2αq

)−(

1−2α

1−2αN

)q′ ( 1−2αq′N

1−2αq′

)}×(

N1−q − N1−q′)−1

, (10)

and can also be expressed as,

HT(q,q′)(πj) =

PN−1(

2 cosh( αq′ ln 22 )

)

(PN−1(2 cosh( α ln 22 )))

q′ −PN−1

(2 cosh( αq ln 2

2 ))

(PN−1(2 cosh( α ln 22 )))

q

}×(

N1−q − N1−q′)−1

. (11)

Equation (11) represents the wavelet (q, q′)-entropy of fractal signals and the expressionPN−1(2 cosh u) in (11) represents a polynomial of order N − 1 which for any N ∈ Z+, is given by:

PN−1(.) = (2 cosh u)N−1 − (N − 2)1!

(2 cosh u)N−3 +(N − 3)(N − 4)

2!(2 cosh u)N−5 − . . . (12)

When q′ → 1, the standard (normalized) wavelet Tsallis q-entropy presented in [25,27,44] is obtainedand with q′ → 1 and q → 1, the wavelet entropy of [24] results. Therefore, with the use of wavelet(q, q′)-entropy, the results previously obtained with wavelet entropy and wavelet Tsallis-q-entropy canbe replicated. The motivation with the use of the two-parameter entropy lies in the increased flexibilityand robustness which can be achieved by the parameter q′.

2.3. The Behaviour of Wavelet (q, q′)-Entropy for Various (q, q′) Pairs

Equation (11) is of particular importance since it allows to construct the so-called entropy planeswhich permit to visualize the behavior of entropies for different q, q′, N and α. A diversity of entropyplanes can be obtained, however, they can be grouped into the following types for the case of fractalsignal classification. Figure 1 shows an entropy plane constructed by setting the parameter q = 7,q′ = 4, using several N and α. Note from the figure that in the limit of α→ 0, entropies are small andappear to be zero, however their true value is:

limα→0H(q,q′)(πj)=

11− Nq−q′ +

11− Nq′−q , (13)

and therefore the well-known Gaussian noise exhibits the smallest (q, q′)-entropy. In general, wheneverq− q′ � 1, fractal signals neighboring α = 0 will also be regarded as processes experiencing smallconstant entropies. Fractal signals with fractality parameter lying outside this interval will have

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a wavelet (q, q′)-entropy that differs from (1 − Nq−q′)−1 + (1 − Nq′−q)−1 and when computed insliding windows they lead to a signal which varies with higher dispersion over time. Therefore,whenever q′ − q � 1 or q − q′ � 1, behavior such as that one obtained in Figure 1 is observed.Note that for this case, the wavelet (q, q′)-entropy behaves as a whitening filter which regards signalsin a specific range as noise. The range of constant entropies can be scaled or comprised according toq and q′, this means that by setting q′ fixed and varying q, a different region of constant entropies isobserved. Note that setting the constant region of wavelet (q, q′)-entropies in the fractality intervalα ∈ (−1, 1) allows to distinguish stationary processes from nonstationary ones simply by observingwhether their entropies are constant or varying and consequently a simple yet powerful classificationtechnique may be devised. A particular advantage of wavelet (q, q′)-entropy over their relatedwavelet q-entropy and wavelet entropy approaches is their ability to analyze the same phenomenausing a different (q, q′) pair. This implies increased analysis flexibility and robustness. Figure 2displays a wavelet entropy plane for fractal signals when q′ < 0 (or when q < 0). In this case, theentropies are similar to the entropies of Figure 1, and experiment roughly two classes of behaviorin the same way as the that observed in Figure 1, the first is associated with constant entropies ina range of the fractality parameters while the second is for varying entropies (for signals with fractalityparameter lying outside the constant range). Unlike the entropies of Figure 1, the entropies observedin Figure 2 exhibit a faster increase outside the interval of constant entropies. The width of constantentropies can be set with parameter q, and in the same way, can be used to classify fractal signalsas stationary or nonstationary. It is important to note that the classification of fractal signals is notlimited to the stationary/nonstationary case but can also be performed for classifying fractal signals asshort-memory and long-memory and also to classify extended fBms from fBms as well, however thepresent contribution only considers the stationary/nonstationary case. Finally, a third way to classifyfractal processes as stationary or nonstationary can also be obtained when the wavelet (q, q′)-entropyhas q � 1 and q → 1. Note that this corresponds to the case of wavelet Tsallis q-entropies and thewavelet entropy plane is identical to the one presented in [14]. Figure 3 displays the wavelet entropiesfor these values of q and q′ along with their contour plots. The wavelet (q, q′)-entropy plane of Figure 3displays constant (maximum) entropies in a range of the fractality parameter and decreasing entropiesoutside this interval. This behavior can also be used for fractal signal classification, however, it ismore restricted in the sense that the range of variation of entropies is small compared to previous(q, q′)-pairs and hence the classification is more complex. Consequently, for the problem of fractalsignal classification behavior such as that observed in Figures 1 or 2 is preferred. In the following,the methodology based on the wavelet (q, q′)-entropy for fractal signal classification is described.

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 8

10

12

14

0

50

100

150

200

Scaling-index α

Length N

(q,q′ )-entrop

y,H

q,q′(π

j)

Wavelet (q, q′)-entropy plane for fractal signals

Figure 1. Wavelet (q, q′)-entropy for fractal signals of parameter α. Parameters q and q′ are set to q = 7and q′ = 4 respectively. Length N is given in powers of 2.

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−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 8

10

12

14

0

1000

2000

3000

4000

Scaling-index α

Length N

(q,q′ )-entrop

y,H

q,q′(π

j)

Wavelet (q, q′)-entropy plane for fractal signals

Figure 2. Wavelet (q, q′)-entropy for fractal signals of parameter α. Parameters q and q′ are set toq = −9 and q′ = −2 respectively. Length N is given in powers of 2.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.58

10

12

14

0.8

0.85

0.9

0.95

1

1.05

Scaling-index α

Length N

(q,q′ )-entrop

y,H

q,q′(π

j)

Wavelet (q, q′)-entropy plane for fractal signals

Figure 3. Wavelet (q, q′)-entropy for 1/ f α signals. Parameters q and q′ are set to q = 7 and q′ = 4.Scaling index range is α ∈ (−1, 1) and the length of the signal ranges in the interval N ∈ (24, 213).

2.4. The Classification of Fractal Signals with Wavelet (q, q′)-Entropy

The theoretical wavelet (q, q′)-entropy planes obtained previously give insight into thecharacteristics of entropies for fractal signals. The behavior of these entropies, for selected values ofq and q′, permit to devise a simple methodology for the classification of fractal signals as stationaryor nonstationary. However, as mentioned above, the fractal signal classification using wavelet(q, q′)-entropies is not only limited to the stationary/nonstationary case but can also be used forthe classification of other classes of fractal processes such as long-memory and short-memory andfor fBm and extended fBm as well. Based on this, Figure 4 displays a generalization of the behaviorobserved in the wavelet (q, q′)-entropies of Figures 1 and 2 and which may permit to classify fractalprocesses as stationary or nonstationary. First, observe that wavelet (q, q′)-entropies are constant overa region −αcoff < α < αcoff and fluctuate for processes whose fractality parameter lies outside thisrange. For the special case in which αcoff = 1, wavelet (q, q′)-entropies distinguish stationary fractalprocesses from nonstationary ones. The classification is thus based on whether the observed signal

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wavelet (q, q′)-entropy fluctuates over a given threshold or not. In order to more efficiently capture thefluctuations of wavelet (q, q′)-entropies, these are computed in sliding windows of the form,

HT(q,q′)(m, W, ∆) = T

{X(tk)Π(

t−m∆W

− 12)

}, (14)

where T {.} is the wavelet (q, q′)-entropy operator over a sample of the signal X(t) and W denotes thewindow size, ∆ the sliding factor and m = 0, 1, 2, . . . , mmax. The use of sliding windows thereforeallows for highlighting the fluctuations of entropies and this variation may be captured with a standarddispersion technique. Since the standard deviation is not robust to peaks and other anomalies withinthe data, the biweight midvariance is used instead since it constitutes a robust estimator of thedispersion within a signal. The biweight midvariance (BW) of a sample {Xi}, i ∈ N is definedaccording to the following relation:

BW =(n′)1/2

[∑|ui |<1 (Xi −M)2(1− u2

i )4]0.5

|∑|ui |<1 (1− u2i )(1− 5u2

i )|(15)

where ui = (Xi −M)/(c(MAD)), c is typically 9, MAD is the usual median absolute deviation andn′ is the number of points which satisfy the |ui < 1| condition. Wavelet (q, q′)-entropy thereforefluctuates for nonstationary processes and remains constant for stationary ones. The estimation ofentropies by sliding windows permits to emphasize further the fluctuations and these fluctuationscan be quantified with the biweight midvariance. In theory, wavelet (q, q′)-entropies for stationaryprocesses are constant, however, in practice, they also fluctuate and a threshold is set in order to classifyfractal processes. Based on this, Figure 5 presents the proposed scheme for classifying fractal signals asstationary or nonstationary using wavelet (q, q′)-entropies.

First, wavelet (q, q′)-entropies are computed in sliding windows and then the fluctuation isestimated using the biweight midvariance technique. Finally, a decision is taken on whether thebiweight midvariance is greater or smaller than a given threshold. In case the biweight midvariance isgreater than the threshold, the process is regarded as nonstationary otherwise the process is stationary.The advantages of the proposed scheme is that overall the process is computationally efficient andtherefore fast for classifying long fractal signals. In the following, a simulation experiment to testthe efficacy of the proposed methodology is presented together with a comparison study against thestandard SSC technique.

0

0.5

1

1.5

·105

−αcoff α = 0 αcoff

Parameter α

(q,q′ )-entrop

y,H

q,q′(π

j)

Wavelet (q, q′)-entropy for fractal signals

Figure 4. Wavelet entropy plane when q < 0 (or q′ < 0). Constant regions are observed inα ∈ (−αcoff, αcoff) and variable regions outside this interval.

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Wavelet(q, q′)-entropy

Biweight mid-variance, BW

BW (HT (πj ,W,∆))

Stationary Nonstationary

BW < thrsh BW > thrsh

Figure 5. Fractal signal classification scheme using wavelet (q, q′)-entropy and the biweightmidvariance technique.

3. Results

3.1. Experimental Results

Figure 5 presented the proposed fractal signal classification scheme based on wavelet(q, q′)-entropies. Note from Figure 5 that a process is regarded as nonstationary whenever BW > thrsh,otherwise, the process is stationary. To get an idea of the fluctuations experimented by the waveletentropies of nonstationary and stationary processes, Figure 6 displays the wavelet (q, q′)-entropies(q = −0.99, q′ = −0.1) of two concatenated fractal signals, the first nonstationary fBm and the secondstationary fGn. Note that the fBm signal experiences a higher fluctuation with peaks while the fGnpresents a lower variation. In biweight midvariance terms, the first presents a higher BW than thesecond and the scheme of Figure 5 may distinguish between these two signals straightforwardly.The fractality parameter of the first is α = 1.05 and for the second α = 0.95. The fractal signalclassification problem has been regarded in the literature as of primary importance, however, all fractalsignal classification techniques experiment errors in the limit of α→ 1. In the following the thresholdthrsh is determined for long and short time series for fixed (q, q′) values and a comparison of theproposed technique with the standard SSC is presented.

0 4000 8000 12000 15000−4−2024

X(t)

XfBm(t)⋃XfGn(t)

0 4000 8000 12000 150000

100

200

300

400

Nonstationary fBm Stationary fGn

Time, t

HT q,q

′(.)

Wavelet (q, q′)-entropy

Figure 6. Classification of signals as stationary or nonstationary. Top plot: concatenated nonstationaryand stationary signal, first half part is a nonstationary signal (fractional Brownian motion (fBm) withα = 1.05) and second half corresponds to a stationary one (fractional Gaussian noise (fGn) withα = 0.95). Bottom plot: wavelet (q, q′)-entropy of concatenated signal computed in sliding windowswith W = 2048, ∆ = 256 and (q, q′) = (−0.99,−0.1).

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3.2. The Threshold for Long and Short Fractal Time Series

In order to obtain the threshold thrsh for classifying fractal signals as stationary or nonstationary,an experimental setup using synthesized fGn and fBm random signals was devised. In fact, for everyfractal signal with parameter α in the range (0, 2), 50 traces were generated and for each, the BWwas estimated and averaged. A plot of α versus the averaged BW can be used to select the requiredthreshold for the given q and q′. The threshold thrsh is determined for both long and short fractaltime series and can be used to classify fractal signals as stationary or nonstationary. Figure 7 displaysthe results of the experimental setup and also the proposed value of thrsh for long and short timeseries. Notice that thrsh should be set to 500 for long time series and thrsh = 100 for short ones. In thefollowing, a comparison of the wavelet (q, q′)-entropy based scheme for signal classification againstthe standard SSC is presented.

thrsh = 500

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

Fractality parameter α

H(q,q

′ )(π

j)

Long fractal times series

thrsh = 100

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

Fractality parameter α

Short fractal time series

Figure 7. Determination of the optimal threshold thrsh for long and short time series. For convenienceonly the range α ∈ (0, 1.2) is shown and every plot was obtained using (q, q′) = (−0.99,−0.1) andW = 256 and ∆ = 64.

3.3. Comparison with the Standard SSC Technique

As previously mentioned, SSC is a well-known technique for classifying fractal signals asstationary or nonstationary. SSC refines the estimation of PSD in the limit of α → 1 and withinthis limit it applies the bdSWV to enhance the estimations. In this interval, SSC basically converts afGn signal into a fBm and an fBm into an extended fBm via a cumulative sum. The next step in theSSC algorithm is to obtain an estimate of H (H). When H ∈ (0, 1), the signal is regarded as stationaryand when H > 1 it is nonstationary. For further details on the SSC and some comparison with thePSD technique refer to the work of Eke [13]. In the following, a comparison study of the SSC andthe proposed technique based on wavelet (q, q′)-entropies is presented. To perform this comparison,exact fractal signals with known fractality parameters are synthesized. The method employed togenerate these fractal signals is known as the Davies and Harte (DHM) technique and the algorithm isdetailed further in [51]. The R package fractal includes the DHM and is used to synthesize fGn, fBmand PPL signals with length N = 214 and N = 211, in the range of the fractality parameter 0 < α < 2and in steps of ∆α = 0.01. For each α, 50 fractal signals are generated and classified using the SSCand the methodology based on wavelet (q, q′)-entropies. Finally, the number of correct classificationsagainst the fractality parameter α is plotted and the accuracy of each methodology compared. The ideabehind the use of PPL signals and the family of fGn/fBm signals stems from the fact that PPL signalsrepresent fractal signals with exact power-law behavior while the fGn/fBm family are approximatelyfractal. Figure 8 displays the results of both techniques (SSC and wavelet (q, q′)-entropies) usingsignals of fGn/fBm type. In this experimental study, wavelet (q, q′)-entropy values were set with(q, q′) = (−0.99,−0.1), W = 256, ∆ = 64 and the threshold with thrsh = 500 for long signals andthrsh = 100 for short ones. Left plot corresponds to long signals (N = 214) while the plot on the rightcorresponds to short-length fGn/fBm signals (N = 211). For long and short signals, the methodology

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based on wavelets tends to have a lower range of misclassifications but has a weaker behavior whenthe fractality parameter approaches the α→ 1 limit.

0 0.5 1 1.5 20

10

20

30

40

50

Parameter α

Cor

rect

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ons

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Figure 8. Classification of synthesized fGn and fBm signals using wavelet (q, q′)-entropy and signalsummation conversion (SSC). Left plot corresponds to signals with length N = 214 and right plot tosignals with N = 211.

Figure 9, presents the classification results of both techniques when using PPL signals of lengthN = 214 (left plot) and N = 211 (right plot). For long PPL signals, the methodology based on wavelet(q, q′)-entropies presents better performance than the SSC technique while for short PPL signalsit is slightly better but overall is statistically similar as the SSC. In summary, the technique basedon wavelet (q, q′)-entropies constitutes a fractal signal classification methodology which presentssimilar classifications as those observed for the SSC, however is more robust in the sense that itpermits not only to classify stationary and nonstationary signals but can also be applied for thelong-memory/short-memory signal classification problem. In addition, the technique based onwavelet (q, q′)-entropies is computationally efficient and can be used to classify long signals fastand accurately.

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Figure 9. Classification of synthesized pure-power-law (PPL) signals as stationary or nonstationaryusing wavelet (q, q′)-entropy and SSC. Left plot corresponds to long signals with length N = 214 andright plot to short-length signals with N = 211.

3.4. Computational Complexity

Wavelet (q, q′)-entropy based signal classification display similar results as that of SSC for longand short fBm/fGn and PPL fractal signals. SSC, however is slower than the proposed technique andnot recommended for long signals. As a matter of fact since the DWT is O(n) and the nonextensiveentropies within the DWT signal take O(log n), then the overall complexity of wavelet (q, q′)-entropyis O(n log n). SSC on the other hand employs a cumulative sum which is O(n2) and an fast Fouriertransform (FFT) technique for the PSD which is O(n log n), therefore, it is extremely slow andprohibitive for long signals.

Entropy 2017, 19, 224 11 of 15

3.5. Application to Financial Time Series

Fractal models have been applied to several financial time series from stock market indices [52]to stock market inefficiency [53]. In the following, classification of stock market indices either asstationary fractal models or nonstationary ones is performed using the method based on wavelet(q, q′)-entropy. For the classification, the parameters are set to (q, q′) = (−0.99,−0.1), W = 256, ∆ = 64and thrsh = 100. The estimated biweight midvariances for these time series are: 120,657.5, 94,943.44,242,874.8 and 109,040.7 for SP&500, Dow Jones, NASDAQ and Nikkei respectively (Figure 10 displaysthe studied time series). As can be noted all these stock market indices are nonstationary and can bemodeled by the fBm signal.

0 5000 10000 150000

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Figure 10. Stock market closing indices. Top left plot displays the S&P index from January 1956 to21 April 2017. Top right plot displays the Dow Jones Industrial Average time series from February1985 to 21 April 2017. Finally bottom left and right plots display the indices for the NASDAQ andNikkei indices.

3.6. Application to Physiological Time Series

Many physiological and biomedical time series are modeled by fractal time series. Gait time series(gait dynamics) are example physiological signals for which fractal models have been applied [54].In the following, the classification of stride interval (gait dynamics) time series using wavelet(q, q′)-entropy is performed. For the classification, the parameters are set to (q, q′) = (−0.99,−0.1),W = 256, ∆ = 64 and thrsh = 100. The time series consist of stride intervals of four young andhealthy subjects walking fast during an hour-long interval. The time series are plotted in Figure 11.The biweight midvariances are 7.653119, 20.82806, 10.71886 and 2.293469 for the si01, si02, si03 andsi04 subjects and clearly indicate that these time series are stationary.

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Figure 11. Stride interval time series from four healthy, young adults walking during a period ofan hour in a 400-m oval.

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4. Discussion

Figures 8 and 9 showed the classification rates of the proposed technique and the SSC forfGn, fBm and PPL signals. Both techniques present advantages and disadvantages. For instance,the technique based on wavelet (q, q′)-entropies have less variation range, however, it presents nocorrect classifications in the limit of α → 1. Overall, both techniques present similar classificationsand can be regarded as statistically similar with respect to the number of correct classifications.Despite these similarities, the wavelet (q, q′)-entropy technique has two additional advantages, it iscomputationally efficient and can be used to classify other types of fractal processes. As a matter of fact,the proposed technique provides a classification in a fraction of a second for long signals while the SSCis slow. Figure 12 displays the average time required to classify a signal as stationary or nonstationaryusing the SSC and the proposed technique. Note that the SSC may require almost an hour to classifya signal with length n = 220 elements. Therefore, although both techniques are statistically similar,the technique based on wavelet (q, q′)-entropies permits to classify signals more rapidly. This accurateand fast classification is important is fields such as computer networking, physics among others.

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Figure 12. Average times for obtaining a classification using the proposed technique based on wavelet(q, q′)-entropies and the SSC. Left plot displays a comparison between these techniques while the ploton the right shows the classification times of SSC.

5. Conclusions

In this article, a methodology for the classification of fractal signals as stationary or nonstationarywas proposed. The proposed methodology is based on two-parameter wavelet entropy which operatesas a preprocessing technique, and the biweight midvariance which measures the fluctuations observedin the wavelet (q, q′) entropies. Theoretical wavelet information planes for these (q, q′) entropies wereobtained and based on these, a classification scheme was devised to regard fractal signals as stationaryor nonstationary. In fact, it was shown that nonstationary fractal signals experience fluctuationsthat are higher than those observed for stationary fractal signals and that this rationale can be usedfor the fractal signal classification problem. Experimental results using synthesized fractal signalsdemonstrate that the technique based on wavelet (q, q′)-entropies is not only comparable in accuracyto the SSC but is also computationally efficient, making it useful for the analysis of long fractal signals.Moreover, it was shown that unlike the SSC, the technique based on wavelet (q, q′)-entropies is moreversatile in the sense that can be used for the classification of other types of fractal processes such aslong-memory and short-memory. Furthermore, by setting q′ → 1 (and also q → 1 and q′ → 1), theresults obtained with wavelet Tsallis q-entropies and wavelet entropies can also be replicated with thewavelet (q, q′)-entropy. Finally, the application of the wavelet (q, q′)-entropy to real physiological andstock market time series indicated that the former are stationary while the latter are best modeled bynonstationary fractal processes.

Acknowledgments: Julio C. Ramírez-Pacheco thanks the support of CONACYT grant for project infrastructureand PROMEP grant. Joaquin Cortez-González thanks the support of PROFOCIE 2014. Joaquin Cortez thanks thesupport of PFCE 2016 and Sonora Institute of Technology through internal funding for research groups (CAs).

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Author Contributions: Julio Ramírez-Pacheco and Joel Trejo-Sánchez conceived the idea. Julio Ramírez-Pacheco,Cortez-González and Ramón Palacio-Cinco designed the numerical experiments. Julio Ramírez-Pacheco andRamón Palacio-Cinco programmed the simulations in R. All authors discussed the numerical results and writethe article. All authors have read and approved the final version of the manuscript.

Conflicts of Interest: The authors declare that there is no conflict of interets regarding the publication of this paper.

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